Forget this idealisation stuff. It has nothing to do with the PI. — StreetlightX

On the contrary. The whole point to this first part of PI seems to be to describe how language is a rule-following activity without having to refer to ideals to account for the rules. I don't want to spoil it for you, but I've been reading ahead, and this will be made quite explicit by the time we reach #100..

In the next section, 60-63, he'll continue his attack on correspondence. He'll compare a language-game in which the named objects are "analysed", to a normal type language-game. Each of the two language-games has its advantages and disadvantages. We cannot say that the "analysed" way provides a better description than the other way, only a different description.

I conclude from this, that we can name what's in the corner of the room as "the broom", or, "the broomstick and the brush which is fitted onto it". One is just as good as the other, but something is lost in each. There is no ideal way of describing the situation, therefore no true correspondence.

Anyway, not a point I really want to follow through on, but this is the second time in this thread where 'things' have been said to stand for words, and it bothers me. — StreetlightX

Why should this bother you? It's how Wittgenstein himself describes a paradigm at #50

We can put it like this: This sample is an instrument of the language used in ascriptions of colour. In this language-game it is not something that is represented, but is a means of representation.—And just this goes for an element in language-game (48) when we name it by uttering the word "R": this gives this object a role in our language game; it is now a means of representation. And to say "If it did not exist, it could have no name" is to say as much and as little as: if this thing did not exist, we could not use it in our language-game.—What looks as if it had to exist, is part of the language. It is a paradigm in our language-game; something with which comparison is made. And this may be an important observation; but it is none the less an observation concerning our language-game—our method of representation.

Notice that the object itself is the means of representation. But that's the nature of a paradigm in this context, whatever it is which is identified as "the paradigm" must represent the name, as the rule instantiated. That's why correspondence is inherently idealistic, as much as we use names to represent things, correct use of names is grounded in an ideal (the paradigm). It might bother you to hear it said that things stand for words, but it doesn't matter because Wittgenstein is rejecting this type of correspondence as unreal anyway. If he stayed on this course investigating correspondence, then when he got to numbers and mathematical names, the paradigms for correct use of these words could only be platonic forms. How could a platonic form be present to us as a paradigm?

A paradigm, whether it is a physical or mental sample, is an ideal. That's why this concept of rules of correspondence needs to be dismissed. It is based in an ideal (correspondence itself is an ideal). The ideal acts as a prejudice, a predetermined way of looking at things (like the example of spontaneous generation), as if the ideal were a real description. But a real description of language never reveals the ideal so the ideal (and therefore correspondence) must be excluded as not part of what language really is.

Isn't the point that they are necessary for the language-game to work? But this is precisely the point: they are necessary, without which the language-game which employs the name in the capacity of a paradigm would not be intelligible. — StreetlightX

I see the point a bit differently. Yes, the paradigm is necessary for such a language-game to work, but only because he has stipulated this by means of the example. It is a presupposition. We have presupposed that there is a language-game which requires rules of correspondence. This language game requires a paradigm.

"An example of something corresponding to the name, and without which
it would have no meaning, is a paradigm that is used in connexion with the name in the language-game."

We can say, that in order for there to be this type of language-game, the game of correspondence, which requires a paradigm, there must actually be a paradigm. However, his demonstrations show that no such paradigms really exist. Therefore we can conclude that there are no such language-games. I believe he is completely rejecting this type of correspondence. I see it as an "ideal correspondence", and correspondence itself has been reduced to being dependent on an ideal. In a few pages from here he will start to discuss the need to reject the inclination to seek such "ideals" in the effort to validate the rules of language-games. So he is here rejecting correspondence because of this "ideal" nature, it cannot be validated as a real language-game.

I agree with unenlightened, it's not philosophy that you are looking for, maybe debating skills or something like that.

When I wrote about the will, or desire, operating within us all, I was actually thinking of the verb.
To will. To want.

Where there's a will there's a way. Angela Merkel also added...' but the will should come from everybody'. The noun is about disposition. Where there is a desire...

As a verb it can express desire, choice. Or a customary habit, natural tendency.
You can call it what you will. You can think of the noun 'will' as you desire.
It might not be the right way, according to some traditionally philosophical way...but it's your way. — Amity

But do you see that what we call "will power" is opposed to this? Will power is what we use to resist the urge to act on what we want or desire. So the two are somewhat opposed to each other and need to be understood by distinct concepts, because a person can get conflicted within, torn up and undecided.

The power of the will, or desire, operates within us all. If there is a lack, then it is more likely to be addressed sympathetically. The causes perhaps being physiological - postnatal depression for example. — Amity

Will and desire are not he same. The will is free, but desire is driven by some underlying condition. The will being free is what allows us to choose a course of action. But in order that the will may have the capacity to choose freely it must also be able to suppress impulsive actions. This is will power. Without will power we'd always be acting impulsively, and never capable of choosing freely. So will has two aspects, one being the capacity to resist actions, this being will power, and the other being the capacity to initiate actions, this being free choice, or free will. Will power is necessary in order that one may exercise freedom of choice.

The difference lies in that we don't need training to be strong-willed. A child is that.
Some might wish to train that out... — Amity

Now I need to ask what you mean by "strong-willed". Assume that the will has two parts, two aspects like I described, will power and free choice. "Strong willed" ought to signify an appropriate balance between the two. Imagine if one's will power was very strong, so much so that the person always resisted actually doing anything. This would not be good. On the other hand, if "strong-willed" meant that the person would persist in a chosen activity, even when that person ought to use will power to resist from that activity, this would be bad as well. So "strong-willed" must mean a balance between will power and following one's free will.

I don’t see that as paradoxical. As I understand it, what he is rejecting is the idea of “an element of reality”. — Fooloso4

I don't see that rejection yet. What I see is that he suggested this idea, that names represent elements of reality, way back in the 40"s. In describing this he saw the need for rules of correspondence. Such rules required a paradigm. The paradigm could not be located so now he's gone back to questioning the idea that names represent elements of reality.

Red refers to a color, that is how we use the name. We do not need the metaphysical framework of elements and complexes to use the word ‘red’ to name something that is red. — Fooloso4

You are just obscuring the problem with this statement. We do not use "red" to name something that is red, we use it to name the colour of that thing. Red is a colour. So we use "red" to refer to a thing, and this thing is a colour. Now we have the metaphysical problem of accounting for the existence of this thing, because the use of "red" implies that there is such a thing, a colour, which is named "red". What we "want" to do here, according to Wittgenstein is just to say that it's a a function of how we use "red" and doing this might avoid the metaphysics. However, the problem is that meaning is use, so if that's how we use "red", that's what "red" means, that there's a thing, a colour called "red". So the attempt to do what we want to do, contradicts itself. The contradiction is put aside by saying "red exists" means that there is something somewhere which is coloured red. But this just brings us back to the problem of the paradigm.

"A name signifies only what is an element of reality [the interlocutor, or his former self] (PI 59)," is not him going back because he is at some "dead end." He is continuing with his analysis of the idea that a name signifies some thing in reality. — Sam26

However, he has just gone through this big discussion concerning rules of correspondence. This came out of the idea that a name signifies an element of reality. There needed to be a rule of correspondence. And this issue has remained unresolved because the paradigm could not be located.. So now he has to go all the way back, to where he was prior to this discussion of rules, to revisit the idea that a name signifies an element of reality, because no rules of correspondence could be validated.

The seeming paradox disappears when the elemental analysis into simples and composites is rejected. — Fooloso4

I don't see how you can say that. The problem, and apparent paradox, is with the supposition "meaning is use". We use the name "red" as if there is something, an element of reality or something like that, which is named as "red". Unless we reject "meaning is use", we cannot reject the "elemental analysis" unless we find some other thing, something other than an element of reality, which "red" refers to.

I didn't assert a contradiction. — MindForged

This is what you said:

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. — MindForged

You are saying that there can be an end point without an order of procedure. That's contradictory, "end" implies order, by definition.. Your dismissal of my argument as "nonsense" relies on the truth of this contradiction. Since it is impossible that a contradiction is true, you need to go back and address my argument properly.

We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. — MindForged

Let me see if I can understand what you're saying. You're saying that we do not produce axioms in geometry for the purpose of measuring things, we do it for some other purpose, maybe just for fun, or some arbitrary, random purpose. Then, voila, it just so happens by some random chance, that the principles of geometry prove to be very useful for measuring objects. Come on, get real.

If you think otherwise, show where measurement appears in the formalism of common geometries. — MindForged

Are you serious? Measurement is everywhere in the formalism of geometry. 360 degrees in a circle is a measurement. Pythagorean theorem is a principle of measurement. I can't understand how you can appear to be so intelligent MindForged, but then fill your posts with such silly and even ridiculous statements.

It doesn't make sense to carry on this discussion because you just defend your position with contradictions and statements of random nonsense. Then you pretend that what I am saying doesn't make any sense in relation to your statements of contradiction and random nonsense.



.

Thanks for this. I have not studied Augustine. I think dividing the human mind into parts - it always seems to be three - is quite problematic. That one follows or rules another... — Amity

As per the discussion in this thread, it is a complicated relationship and we cannot say that one follows or rules the other. This is why we can talk about things like training the will in good habits, and training to be strong willed.

Some think we should do away with the concept of willpower altogether. Instead of focusing on it, we should be examining the power of will. Basically, I think we give up on projects that don't engage us. — Amity

I don't understand this statement. What would be the difference between "will power" and "the power of the will"?

§58
If the meaning of a word is not tied to an object, paradigm, sample, memory, or any other object (mental or otherwise), then it seems to follow that the meaning of "X exists," is derived in another way. In particular, meaning is derived how it is used in social contexts. So, "X exists," if it is to mean anything, means, there is such-and-such a use for the word. Although as Wittgenstein points out this is senseless.

We could extend this to the proposition that "God exists," which does not derive meaning from whether or not the thing associated with the concept has an instance in reality, but how we use the concept in a variety of social contexts. We should not think that a name is only meant to be some element of reality (PI 59). — Sam26

How do you relate #59 in this way? It appears to me like "meaning is use" has met the paradox of 58. We want to say "red exists" means that the word red has meaning, rather than that there is an existing thing called "red". However, since meaning is use, and we use "red exists" to say that there is something, a colour called "red", we cannot do what we want to do, the attempt contradicts itself. So it appears to me, like he has met this dead end, this paradox at 58, so he goes all the way back to the proposition "A name signifies only what is an element of reality" at 59, to get a fresh start, from a new perspective.

Wittgenstein states that we want to take "Red exists" as "'Red' has a meaning", and "Red does not exist" as "'Red' has no meaning". — Luke

The problem is that meaning is use . And, we use "red" in this way, as if the word refers to a thing, "red exists", "red is a colour", etc.. So if we claim "red exists" doesn't really say anything about a thing named red, it only says something about how we use the word, then we must look to the use of the word for its meaning and we find that we use the word as if there is something called "red" which exists, So that's what "red exists" actually means. This is why "what we really want is simply to take "Red exists" as the statement: the word 'red' has a meaning", ends up contradicting itself in the attempt.

He seems to propose, at the end of 58, that what "red exists" really means is that there is something existing which has the color red. And when he suggests "what has that colour" is not a physical object, he must be referring back to the "mind's eye", or memory, at 57.

However, I would say that it's doubtful that he has proved at 55-57 that for "red" to have meaning requires that there is something which has that colour. It appears to me that the word "red" could still have meaning when there is no red physical object, nor such a colour in anyone's mind, as this is the case when we create imaginary scenarios. So one might say "red is a colour", while there is no red physical object, nor the image of a red colour in any mind, and "red" would have meaning in this imaginary scenario. This is demonstrated by Fooloso4's example, "greige" is a colour. In this case "greige" has meaning, as a colour, and there is nothing, in the physical, nor the mind, which has that coulour. The word "greige" receives its meaning from the context of use, "is a colour"

.

Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. — MindForged

Right, you think that there could be an "end point" without an order. You really like to argue by way of contradiction, don't you?

For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over. — MindForged

Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion.

And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning. — MindForged

Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question.

So, what kind of infinite thing (infinity) do you think you could observe?

Word game. By "reality" I meant being actual. You've already said you don't think this is possible, — MindForged

That's what I meant, "actual". If you saw my discussion with aletheist, you'd see that. For Aristotle, ideas, concepts, have actual existence, actualized by the human mind. This is the argument he uses against Platonic idealism. Ideas cannot be eternal, because only actual things can be eternal, and ideas are only given actual existence by the human mind, so they have a beginning and are therefore not eternal.

My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing. — MindForged

More of the same, nonsense. The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter.

I think that in most forms of idealism the mind doesn't have a spatial existence, so it doesn't make sense to ask where is the mind, like it doesn't make sense to ask where is the future, or where is the past. You are attempting to produce a spatial context where there is none.

How does the idea of a tree become a real tree? where is it? where are we? — Jamesk

It is an issue of temporal continuity At each moment in time, what exists at one moment is replaced with what exists at the next moment. "Real tree" implies that the thing observed as a tree is something with temporal extension, continuous existence in time, rather than just a flash of existence at a particular moment. But only a mind with a memory of past moments, establishing a relation between these moments can synthesize the reality of a tree with temporal extension. So the reality of the tree, with temporal extension, is an ideal created by that mind.

Moving along, at 58 Wittgenstein exposes a complex metaphysical problem. He has already distinguished between having a physical colour sample, and the memory of a colour, (as comparable to having a physical sample). Neither one of these suffices to account for the meaning which the name has. Now, at 58 he discusses the name directly, "red" for example.

He proposes that "red exists" is meaningless. This is because that usage makes it appear like there is a thing named as "red", which exists. In reality there is no such thing (as demonstrated 55-57). So there is just the use of the word "red". That there is no such thing as that which is named by "red" is apparent from the above: it is not represented by the physical sample, nor is it represented by the memory. So "red exists" is just a misleading way of saying that the word "red" has a meaning. And "meaning" is a representation of the use of the word in language.

At the end of 58 he discloses a slight problem with this approach. He admits that "red exists" doesn't really say "the word red has meaning", but this is what it would have to say if it meant anything. What it really says, (according to how it is used), is that there is something named "red", which exists. However Wittgenstein has demonstrated that this is meaningless because there is no such thing as the thing which is called "red". And so he concludes that if "red exists" is to mean anything, it must mean that the word red has meaning.

You can see how it "contradicts itself in the attempt". The way "red exists" is used, is to signify that there is something called "red", which exists. But, there is no such thing as what is referred to by "red", only the meaning or use of "red". I believe it's a sort of paradox, and we might say that using the word "red" in this way creates the illusion that there is something named by "red", and, since meaning is use, the thing referred to, in this usage, must be in some way real because that's the meaning the usage has given it. So Witty's prior demonstration (55-57) indicates that there cannot be anything referred to by the word "red". However, the word is used as if there is such a thing. And, since meaning is use, the meaning is that there is such a thing. But to say that there is such a thing is to say something meaningless.

Not an infinite regress, no. — Isaac

I don't see how you can deny an infinite regress. If your description of "learning a rule" includes that the person already knows a rule, then unless you can account for this already known rule in terms other than as "a rule", infinite regress is implied, and it is false to say that the description is "complete".

It's not that "learning a rule requires that one already know a rule" it's that all the rules Wittgenstein is interested in here, are of that sort. The description of our first acquisition of rules, our first tentative steps, is a matter of of child psychology, not philosophy of language. — Isaac

So what you are saying here is that Wittgenstein is not describing what "learning a rule" is. He is describing what learning a particular sort of rule is. Maybe we could say that he is describing learning the sort of rules which apply to games. And, to learn this sort of rule requires that one already knows another sort of rule. This is good, but if he later attempts to define "rule" such that all rules are of the sort he is describing (circumscribe the region as per #3), then this other sort, the sort which is a prerequisite to the sort he is describing, needs to be accounted for in terms other than as a "rules", or else the definition is faulty as per #3.

It is sufficient for this investigation, that Wittgenstein's "close examination" has shown no 'rule of rules', his examples have pointed fairly conclusively to the rule-following being situated firmly (and complexly) within the social context. Somewhere in the millions of interactions emerges the rule, just like somehow in the millions of interactions between air molecules emerges the weather patterns. — Isaac

Right, from somewhere within those dusty rags, the mouse "emerges". Tell me another one bro. Isn't the "close examination" intended to get beyond this sort of thinking?

If you don’t know what color to paint then the name greige is meaningless. — Fooloso4

All I can do is repeat. If you do not know which colour "greige" refers to, but you know that it refers to a colour, then the name is not meaningless to you.

Knowing that it is a color is meaningless for the purpose of painting or picking out a fabric or whatever else you might do with a specific color if you don’t know what color it is. — Fooloso4

But it doesn't make sense to say that if a word is useless for some particular purpose it is therefore meaningless. There are many, many, words which are useless for the purpose of picking out a paint or a fabric, but this does not make them meaningless. You are simply declaring that if the person cannot carry out a very specific task related to the word "greige", get a greige coloured paint, then the word "greige" is meaningless to that person. When in reality the word does have meaning to that person because the person knows that it refers to a colour. There is probably hundreds of colour names. The vast majority of them I am incapable of picking out the corresponding colour. However, I would recognize very many of them as colour names. It's nonsense to suggest that just because I cannot identify the corresponding colour, the name is therefore meaningless to me.

As far as my understanding goes, Socrates is not saying spirit is always an ally of reason. Instead, he is giving an example where reason, being firmly in control, may ally with spirit to control desire. After all, shouldn't reason propel a man to feel anger for being a slave to his desires? Thus there are cases in which spirit can aid reason. A man's passions are a powerful thing, and if guided can lead a man to greatness. If they are not, they may lead him to ruin. If they are denied or suppressed, they will surely return with a vengeance. — Tzeentch

Yes, I believe this is a good description of the tripartite soul. The spirit (sometimes translated as ambition), is like a medium between the mind and the body. It is how the two distinct categories, mind and body interact. Through the means of spirit or ambition, the mind may have control over the body. But in an ill-disposed, poorly tempered person, the opposite may be the case, and this is a corruption of the soul. It is suggested that we might be able to culture the proper balance, and in The Republic the suggested balance between training in gymnastic and music appears to be prominent towards this end.

Akrasia (/əˈkreɪziə/; Greek ἀκρασία, "lacking command"), occasionally transliterated as acrasia or Anglicised as acrasy or acracy, is described as a lack of self-control or the state of acting against one's better judgment.[1] The adjectival form is "akratic". — Amity

Augustine considered this problem quite extensively. How is it possible that one can know what is good, and even decide to do the good action, yet still proceed to do the contrary? I believe that this is the root of his division of the human mind into three parts, memory, intellect, and will. It is an extension of Plato's tripartite soul. With this division, the will does not necessarily follow what the intellect. Later, Aquinas discusses the relation between intellect and will. Although the will is generally seen to follow the intellect, in the absolute sense will is prior to intellect. This is how we can uphold Augustine's conception of free will.

He starts the quoted phrase by saying: “When we forget which colour this is the name of …”. What is not remembered is what the color “greige” means, that is, what color it is. We might remember the color of the foyer but not remember that the color is called greige. — Fooloso4

No, "we forget which colour this is the name of" says that we forget the colour, not that we forget the name.

So if someone asked you to paint the bedroom greige it would have no meaning. — Fooloso4

Not necessarily, because you might still remember that "greige" refers to a colour, but just not remember what colour it is. In this case, the loss of meaning of "greige" would not be complete, or absolute. The word would still have some meaning, it is understood to refer to a colour.

It is the situation that is comparable. Suppose the person who wanted you to paint the room found a color swatch and wanted the room painted that color, but could not find the swatch to show you. The paradigm, in this case the swatch, is lost. It would be meaningless to ask that the room be painted the color of the swatch if there is no swatch. — Fooloso4

OK, but notice that when the colour swatch is lost, the name of the colour still has some meaning. The person recognizes the name as being the name of a colour, but just doesn't have any way of knowing exactly what colour it is.

This does not mean that the object cannot be a paradigm but that a paradigm is not necessary when the connection between the name and the thing named has been made. When the name would have no meaning for someone without an example, a paradigm is used, an example. That example might be an object, but if one already knows that this thing is called “xyz” then “xyz” still has meaning even without the presence of an object. — Fooloso4

I think that what is demonstrated is that the meaning of the name is not rooted in the paradigm (as physical example) at all. A paradigm is not necessary for the word to have meaning. The person forgets what colour "greige" refers to, but the name still has meaning as signifying "a colour". The colour swatch is lost, and the name still has meaning as signifying "a colour". The existence of a paradigm (physical or remembered example) is not necessary for a name to have meaning.

In general, the meaning of a word is determined by its use: — Fooloso4

Right, the meaning of a word is determined by its use. Is this something distinct from "a paradigm"? If so, then this would mean that the idea that the meaning of a word is determined by a paradigm can't be right. But we can create compatibility if the paradigm is somehow a paradigm of use.

If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right? — aletheist

"Infinite" is a description of the numbers, as such it is qualitative, not quantitative. Mathematics cannot deal with the concept of "infinite" which is a description of mathematical objects made from outside the principles of mathematics, because it is not a mathematical principle. That's the problem here. The point being that whatever category you put the mathematical objects into, the descriptive term "infinite" is of another category, as the difference between the territory and the map, one being the object, the other a description of the object. The problem occurs when we attempt to make "infinite" a mathematical object.

Where on earth have I ever suggested that ideas are not real? — aletheist

You insistently claim that numbers have no actual existence. The #1 definition of "real" in my OED is actually existing. So I concluded that you do not believe numbers to be real. You insist that numbers cannot interact with things in the world. Now you claim that ideas are real. I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. What sense is there in this, to allow for something real, which cannot interact with anything else in the world? So I haven't any idea what you might mean by "real" now because you seem to be claiming that there are real things which cannot in any way interact with the world we sense.

... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible. — aletheist

I don't see how "possibility" is at all relevant in your reality. It cannot interact with the actual world, as a distinct mode of being, so how could it be relevant? And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed.

There are a couple of issues with this; firstly, why would you be so concerned that a method provide principles for strictly judging correct from incorrect? I mean, what's the goal here. Is it just so that we can enjoy policing language users who've 'got it wrong'? What use would we put such a rule to if we found one? — Isaac

Isn't that what a "rule" is though, a principle for judging correct from incorrect? If we are here concerned with "rules", but as you suggest, we are not concerned with principles for judging correct from incorrect, what could we possibly be talking about?

That is the full description of their 'learning a rule'. — Isaac

If the full description of "learning a rule" requires that one already knows a rule, and this produces an infinite regress, then obviously this description is faulty.

So what do you think we can conclude from this? If describing language use as a case of rule-following results in such an infinite regress of needing to know a rule in order to learn a rule, ought we not conclude that this is an inadequate description? Doesn't this demonstration prove to you, as it does to me, that it is impossible that language use is a simple case of rule-following?

Consider that we have to account for the creation of rules as well, and this will come up later. It is impossible that creating rules is a rule-following activity or else we have the infinite regress. So if his intent were to make the description of language as a rule-following activity, into a true description, then the rule-creating activity would have to be excluded, as not part of language. Remember #3, we can make a description appropriate by circumscribing the region.

3. Augustine, we might say, does describe a system of communication; only not everything that we call language is this system. And one has to say this in many cases where the question arises "Is this an appropriate description or not?" The answer is: "Yes, it is appropriate, but only for this narrowly circumscribed region, not for the whole of what you were claiming to describe." — Wittgenstein

Which leads me to the third point. The reason why Wittgenstein writes the way he does is widely considered to be his solution to this very problem. He's trying to get us to see something which actually cannot be said, it can only be shown. He's not constructing a watertight argument in logic, because there is no such thing. He's pointing out things which should lead us to 'see' what he's trying to show. Rather like someone trying to point out the beauty of a sunset by gesture alone, it's not going to work unless you're looking where he's looking. — Isaac

Yes, this is good, but it's also a big problem for interpretation. To "get it" is not necessarily to understand the particular words, but to see what's being pointing at, beyond the particular words, the bigger picture of meaning. So let's say it's like you say, similar to pointing to the beauty of a sunset. Someone points and says look at that isn't it beautiful. One person might see this or that colour in the sky, another something on the horizon, another a pattern in the clouds (interpretation of art is similar to this), but what the person is really pointing to is the whole scene, as beautiful. The beauty is in the whole scene, not in any particular part. I think meaning is like this, how we see the whole scene, not any specific part.

Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists. — aletheist

You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true.

However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. — MindForged

Actually, that is what is mistaken. In the case of the reals between 0 and 1, it is quite obvious that 1, like 0, is a defining point, and therefore a beginning point rather than an end point. What is claimed is that there is an endless number of points between 0 and 1. It is impossible that there could be a direction of procedure, because if we started at zero, and tried to progress toward 1 as an ending point, it is impossible to name the first real number after zero. Any named number after 0 would have more numbers between it and 0, and we'd have to turn around and go back, heading away from 1. Therefore, in this instance it is false to represent 1 as an "end point". There are two defined start points, 0 and 1, with an infinity of points between them. No end point.

didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? — MindForged

I guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite.

It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. — MindForged

I know, I believe I already described to you how definitions are arbitrary. But those definitions which demonstrate a true correspondence are considered to be true definitions. So if we observe that the clear sky is always a similar colour, and we name this colour as "blue", so that everyone calls this colour blue, then we can define "blue" as "the colour of the sky". There is true correspondence because that is how people use the word "blue". But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence.

Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible — MindForged

Why is this "worse"? Time and space are purely conceptual, just like numbers. Numbers are conceptual and infinite. If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? A conclusion reflects its premises. The premise is that numbers are infinite. If time and space are concepts created from numbers they will reflect this infinity. Unless we allow that time and space are concepts created by something other than mathematics, they will necessarily be infinite. If time and space are other than mathematical, what would be the basis of these concepts, observation? Observations are necessarily finite. Therefore we have an incompatibility between the concepts of space and time which are consistent with mathematics, and the concepts of space and time which are consistent with observations. This has manifested as Zeno's paradoxes.

I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object. — MindForged

The problem is that things unobserved do not enter into conceptions produced from observations So, even if there is a real thing out there, like space or time, which is truly infinite, the limitations of our senses deny us the capacity to observe the infinity of this thing. That's the classical, or colloquial understanding of "infinite", that it's impossible for the human being to observe. Let's assume that space and time are infinite, as the mathematical conceptions tell us, but our observations are incapable of corroborating this due to the limitations of our senses. Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? Do we face the fact that our observations are limited, and therefore fail us in this realm, and maintain a pure infinite in our concepts of space and time, or do we denigrate the pure infinite concept, and produce a new concept of "infinite" which is more consistent with our faulty observations? The latter is what the logicians have done, and what you seem to insist was the right thing.

You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. — MindForged

What you're not respecting, is that for Aristotle ideas are part of reality. He was a student of Plato and was well trained in an ontology that holds ideas as real. To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. Infinity, as well as mathematical ideas are very real for both Plato and Aristotle, so placing "infinity" into the category of ideal is not partitioning it away from reality.

After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have. — MindForged

You seem to be missing the point. I agree with what you have described here, a possibility is defined by actuality, what actually is. This is a specific possibility, it is only correctly "a possibility" if the actuality permits, otherwise it's impossible. Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible?

I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me? — MindForged

Yes, I see why my view of geometry makes no sense to you. You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that?

Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary. — aletheist

Actually you seem to have misunderstood what I meant. I didn't mean mathematics is necessary for building all things, but for some things. So my argument remains the same. Of these things which mathematics is necessary to build, the mathematics must somehow interact with things in order that these things get built.

Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary. — aletheist

I don't see how my computer could have been built without mathematics. Regardless, let's just say that mathematics is useful for building things, as you say. How could mathematics be useful in building things unless it somehow interacted with things? You might say that the human being is a medium between the thing built and the mathematics, but the human being is also a thing, and the mathematics must interact with that thing in order for it to build the things which the mathematics is useful for. So the mathematics still interacts with things, even though the human being, as a thing is a medium between the mathematics and the thing built..

This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real. — aletheist

I never made any such claim so instead of addressing my concerns you are just changing the subject.

My first challenge to you would be to find other philosophers who see it the way you do, no interpretation is a matter one person's view, as if you can simply choose any interpretation you want. So why don't you find other philosophers who view rule-following the way you do, and present the argument. — Sam26

So far I am in complete agreement with StreetlightX's interpretation, and I seem to be in agreement with Isaac. I have been pretty much in agreement with Luke up until now, but this particular section we're on now appears to be difficult.

One of the things you seem to have a hard time with, is that rule-following is intrinsic to the actions associated with linguistic activities. — Sam26

You keep asserting this, but I've followed the text quite closely and Wittgenstein has yet to make such a claim. Whether this is what you or I believe is irrelevant, we are interpreting the text for what it says.

My second challenge to you is to explain Wittgenstein's rule-following argument as you understand it, whether you disagree with it or not. Explain it like you were explaining it to someone who never read Wittgenstein, and use supporting paragraphs. — Sam26

At this point in the text, where we're at, Wittgenstein has not yet produced a "rule-following argument". It actually doesn't seem to be the type of book which proceeds by arguments, more like he makes various points through multiple examples. If you could point me to such an argument, I will offer an interpretation of the passage. Otherwise you might just follow the thread and when this so-called rule-following argument comes up I'm sure we will discuss it.

Right, if there were no sample that could be used as a paradigm then there is no way to settle whether one remembers the color correctly. It is possible that the sample has darkened but it is also possible that one does not remember the color correctly.
Such indeterminacy or uncertainty is not something Wittgenstein is attempting to overcome. See below regarding rules. — Fooloso4

OK, so I think we agree on that, but how would you interpret this passage at the end of 57:

When we forget which colour this is the name of, it loses its meaning for us; that is, we are no longer able to play a particular language-game with it. And the situation then is comparable with that in which we have lost a paradigm which was an instrument of our language, ...

Is he saying that what was referred to as a paradigm, in 55, "something corresponding to the name, and without which it would have no meaning", actually exists in memory? Or what does he mean by "comparable" with a paradigm? He seems to say at 55 that an object cannot be such a paradigm because the name can still have meaning without the object, but now he says that the name cannot have meaning without the memory. So isn't it the memory which fulfills the conditions of "a paradigm" as stated at 55, something corresponding to the name, without which the name would have no meaning?

If the paradigm is within the memory, and the memory is unreliable, then how could we ever know the correct use of the word?

An example of a useless erection is when you awake and gather the morning wood for the fire, erect that morning wood so it will rage when lit, but others have no immediate interest in it, so instead of it casting copiuos emissions, it just wanes, sputters, and sits uselessly. — Hanover

I suppose it would be a moron who would spend multi-billions of dollars on such an erection. The question I guess, is why wouldn't the border wall wane, sputter, and sit uselessly, as a useless erection?

What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else. — aletheist

How is it that mathematics is necessary for building things, yet numbers do not interact with anything? That's the nonsensical claim, that numbers do not interact with anything. I suppose engineering could be done without numbers? And if numbers are necessary, how so if they don't interact with anything? Obviously numbers interact with things or else they could not be necessary for building things.

False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum. — aletheist

Of course they're not measurable shades if they're not actual shades, only potential shades. They are simply imaginary, so of course they cannot be measured. Is this how you conceive of the continuum as well? Is it simply imaginary as your example seems to indicate? I think it's purely imaginary, don't you?

By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero. — aletheist

This is what is nonsense. Your one-dimensional infinitesimal is just a short line. You arbitrarily claim that its length is less than any assignable value, but there is no such limit to our capacity to assign a length value because numbers are infinite. So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit.

There is a reason why it is in parentheses. The next paragraph begins -

“One might, of course, object at once …”

And concludes:

“An example of something corresponding to the name, and without which it would have no meaning, is a paradigm that is used in connexion with the name in the language-game."

The example serves as a paradigm for something that corresponds to the name. — Fooloso4

OK, I see that, but then he proceeds with "what if no such sample is a part of the language..." and proceeds from here. So it is still undetermined as to the criterion for "correct".

No serious student of Wittgenstein, if even they disagree with Wittgenstein, would deny many of these points being made about rule-following. There may be disagreements, but most of this is understood by those who study Wittgenstein in a serious way. — Sam26

I don't disagree with Wittgenstein on these points, but I disagree with your interpretation. So you have a specific interpretation, which may or may not, to some extent, be shared by other serious scholars. And so you might think that I disagree with Wittgenstein, simply because the interpretation I make, and which I agree with in principle, disagrees with yours.

The one thing I can say about MU is that he keeps his cool about all of this. In that sense he is better than me. The Marine in me wants to take the person and kick them in the ass, which would solve nothing. I think it comes down to this, you either see it or you don't. If you don't see it, fine, just move on. Now, if I can take my own advice, I'd be doing well. — Sam26

So why is it a case of "you must see it my way or else you don't see it at all"? It really does not come down to "I must see it your way, or else I don't see it". What makes your way the correct way?

My point is that if you think you're going to get anywhere with your explanations you're living in a dream world. The only reason I see to answering some of his questions is to help others who may also be confused. — Sam26

Hey Sam26, in case you haven't noticed, although I still have a lot of confusion, my perspective on this material has changed a lot since we first engaged years ago. Wouldn't that indicate that your explanations have helped me? Allow me to thank you for that. Thanks, sincerely. Now what about you? Do you steadfastly maintain the same interpretation you made on your first reading, or do you proceed with an open mind, hoping to learn something new every day?

Again, where is the problem if that "something" is mathematical--i.e., hypothetical--rather than actual? Are you claiming that reality is limited to that which is specific, definite, and measurable? If so, on what grounds? — aletheist

You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? What nonsense is that? Mathematical objects exist as specific definite things. That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. That's why the attempt by speculative logicians and mathematicians to bring "infinite" into the realm of mathematical objects is doomed to failure as inherently contradictory.

A color is a quality, so its mode of being is that of possibility. Between any two "measurable" shades of red (for example)--e.g., identified by RGB hexidecimal code or electromagnetic wavelength to an arbitrary degree of precision--there are intermediate shades beyond all multitude. All of them are real, regardless of whether they ever exist by being instantiated in actual concrete particulars. — aletheist

Each of those shades of colour is measurable though. You have defined the infinitesimal as having an immeasurable dimensionality, yet still having a dimensionality. Since dimensionality constitutes being measurable, this is like saying that infinitesimals have something measurable (dimensionality), which cannot be measured. That's blatant contradiction.

I thought it was obvious in context that I was talking about a one-dimensional infinitesimal for the sake of conceptual simplicity. Its "length" is non-zero, yet smaller than any assignable value. As such, how could we measure it, even in principle? — aletheist

The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal".

I don't disagree with this, I only mean to emphasise that Wittgenstein is not saying that 'rule' should mean something and then proceeding to see what meets this definition, but rather working the other way around. — Isaac

So we're in good agreement here.

I felt like your point was implying that some situation might exist where we say a game is being played according to a rule, but which, on investigation we find it is not (ie your comment about the assertion needing to be justified). I was trying to get across that Wittgenstein is simply accepting that whatever we say is being played according to a rule is part of the definition of 'rule' so he's not (at this stage) undertaking an analysis of whether people are right to say that a game is being played according to a rule. — Isaac

This is where we may have a problem, and fall out of agreement. I really believe that there are difficulties with this method, Platonic dialectics, which Witty will expose with his discussion of what is a game. The same word will be used in a wide variety of ways with some sort of thread of relation. We, as human beings have an intuitive inclination to judge the "correct" way. But the method has no means of determining the correct way. According to the principles of this method, there is no such thing as the correct way, there is just a variety of different ways, somehow related, which all need to be examined. So the common notion of "a rule" turns out to be incompatible with this method. There is no rule as to what a game is. And even the word "rule" refers to many different things. This ought to leave us at a loss as to how to ground the idea of correct and incorrect, which is why Plato turned to "the good". Whatever form of grounding one chooses, it may appear to be an arbitrary choice. So we will likely fall out of agreement on this because I think that the method provides no principles for judging correct and incorrect .

An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality. — aletheist

You don't see this as a problem? Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. What could this possibly mean, other than something contradictory? It doesn't have any measurable colour but it has real colour.

Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit. — aletheist

But a line is specifically one dimensional, and an infinitesimal is not. So if you zoomed in on an infinitesimal why would you see it as a one dimensional line rather than as three dimensional, four dimensional, or even an infinity of dimensions for that matter? If it might be an infinity of dimensions, then the purpose of the infinitesimal is self-defeating.

Infinitesimal point" is self-contradictory--points are, by definition, dimensionless and indivisible; infinitesimals are, by definition, dimensional and potentially divisible without limit. As I have stated repeatedly, in my view an infinitesimal is not a discrete unit of any kind, and a continuum is not composed of infintesimals. — aletheist

That's exactly the problem with infinitesimals, their dimensionality is ambiguous. If it is not a discrete unit in any way, then it has no form and therefore no specific dimensionality. However, you say that it is not dimensionless. Therefore it has dimension but its dimensionality is completely undefined and ambiguous. But dimensionality is purely conceptual, and must be defined. Now we have an infinitesimal which is defined as having no specific dimensions yet it is not non-dimensional. Therefore it is either completely irrelevant to our conceptualization of dimensionality, or it occupies some vague, ambiguous, undefined position within our conceptualization of dimensionality which could only serve to mislead us.

No, that presumes that the definition of "rule" exists prior to our investigation and we are demonstrating that the thing we have identified belongs to that definition. That's not what's happening here. Wittgenstein is saying "let's call the paradigm that looks like it is needed to explain this broadly similar collection of behaviours a 'rule'", then look at some examples to see how it varies. You're starting from the premise that a 'rule' is a thing of universally fixed and agreed on definition and the game is to try and see if what Wittgenstein describes is such a thing. You may play that game with your definition of 'rule' but it's not a game I care to play. — Isaac

I'm fine with this, but then it makes no sense to say, as you did, that if you are able to respond when someone asks you to "fetch a red apple" then you know the rule, because this also presumes a definition of "rule". You cannot exclude my definition of "rule" because it is presumptuous, and then proceed with your own presumed definition.

This is an important point demonstrated in Plato's Theaetetus, and it is evident that Wittgenstein is familiar with this work. The participants in the dialogue proceeded toward defining "knowledge". Each definition which they tried out, based on examples of how knowledge appeared to exist, proved to be faulty in their enquiry. So they failed. At the end, they realized that they had failed because they had a preconceived idea of knowledge, as excluding the possibility of falsity, or mistake, and this preconceived idea was wrong in comparison with how knowledge actually exists. So this presumption that knowledge had to exclude mistake or falsity prevented them from being able to say that any of the descriptions of knowledge, which they took from real examples, qualified as "knowledge" according to that preconceived notion. Knowledge in the existent examples didn't have the capacity to exclude the possibility of mistake or falsity.

So here we have a very similar issue. If we have a preconceived idea of what "following a rule" is, this will stymie our attempt to look at the real world instances and determine what "following a rule" really is. On this basis, I disagree that Wittgenstein is saying "let's call the paradigm that looks like it is needed to explain this broadly similar collection of behaviours a 'rule'", because this is to proceed with a preconceived idea as to what a rule is. What Wittgenstein did, in fact say at 53 is: "Let us recall the kinds of case where we say that a game is played according to a definite rule." Then he goes on at 54 to explain those kinds of cases.

Therefore he has not made the generalization which you claim. He has not claimed to call "this broadly similar collection of behaviours" is a "rule", he has only described a variety of different behaviours in which we say that a game is being played according to a rule.

How would you know when you find such a thing? — Isaac

I think that this is a very good point, and it is the question which came to my mind after reading Plato's Theaetetus. If we proceed to define "knowledge" by looking for instances of knowledge, and then producing a definition of knowledge from that, how would we know in the first place which of the things we are looking at, are knowledge. How can we find what we are looking for, if we proceed with no idea of how to identify it. But this is the Platonic method, it's called Platonic dialectics, and it bears a strong resemblance to Wittgenstein's method. The method is to examine the sorts of things which are referred to by a particular word (usage of that word) and produce a definition of the word from the way it is used. We know which of the things we look at are the ones we are looking for by the use of the word. If the thing is called "knowledge", then it is an example of the thing we are seeking to define. So in the example of the Theaetetus, we would approach "knowledge" without any preconceived notions of what "knowledge" ought to refer to, then examine all sorts of instances where the word is used, and develop an understanding of what "knowledge" means from that. So we do have a means of identifying the thing which we are trying to define, and that is the usage of the word. I think Wittgenstein is saying we ought to approach "rule" in this way. And soon he will demonstrate the profound difficulties with Platonic dialectics when he asks what sort of thing is a "game", and mentions a vast variety of different things referred to by that word. Now he has mentioned a variety of different things referred to as "playing a game according to a particular rule".

Two different senses in which one can follow a rule. The first means to understand what the rule is. You have indicated that you can follow the rule to fetch a red apple but choose not to. Do you follow? — Fooloso4

I follow, but clearly the point Wittgenstein is making at 54 is that there are different senses of "following a rule". On what basis would you choose one over the other as the correct sense?

Are you referring to the statement in §55 in quotes? If so, that is not Wittgenstein's position, it is one that is said that he rejects. — Fooloso4

All I see is that it is stated, I do not see him rejecting it.

Once the connection between the name and the thing named is made the paradigm is no longer needed, ... — Fooloso4

I do not see where Wittgenstein makes this claim. In fact, if this were the case, then we'd have to rely on memory. But he explicitly rejects a reliance on memory at 56.

Then what do you think it refers to? — Fooloso4

Don't ask me, I didn't write the book, and as far as I can tell he hasn't elucidated this yet. Maybe he likes Platonic realism in which the example (paradigm) exists in some eternal Platonic realm, or maybe he'll put forth some other platform. I don't know.

Continuum to me implies a smooth transition between two points. — TheMadFool

Continuum is uninterrupted. "Two points" implies two distinct places and therefore a boundary which separates them.

However, we can choose arbitrarily small units and do any measurement. What I mean is we can make any quanitification arbitrarily smooth depending on our needs for accuracy or whatever. — TheMadFool

So I assume that the "smooth transition" you refer to is arbitrary and not real?

You misread what you quoted. I said the mathematical conception has no contradictions, I didn't say it was identical to the colloquial definition. The colloquial understanding of infinity includes, for example, a notion of unboundedness. And yet we know infinities are in some sense bounded, and people will readily admit that the real between 0 and 1 are infinite despite that clearly being a bounded array of values. That's just an obvious case of a colloquial, folk conception being contradictory and hence the need for a formal understanding which we got from mathematics. — MindForged

I agree that the mathematical conception has no inherent contradiction. But the only sense in which "an infinity" is bounded is by the terms of its definition. All infinites which we speak of are bounded by the context in which the word is used. If someone mentions an infinity of a particular item, then the infinity is bounded, defined as consisting of only this item. Likewise if we are talking about an infinity of real numbers between 0 and 1, the infinity is bounded, limited by those terms. However, we are not discussing particular infinities here, which may be understood as particular (though imaginary) objects, we are discussing the concept of "infinite". We are not discussing conceptual entities which are said to be infinite, we are discussing what it means to be infinite. The reals between 0 and 1 is a conceptual entity which is said to be infinite. We are asking what does it mean when we say that this is infinite.

We don't derive from observation that the infinite is relegated to ideas — MindForged

This is false. Anytime "infinite" is used to refer to something boundless, or endless, it refers to something made up by the mind, something imaginary or conceptual. We do not ever observe with our senses anything which is boundless or endless, because the capacities of our senses are limited and could not observe such a thing. Since the capacities of our senses are finite we know that anything which is said to be infinite is a creation of our minds, it is conceptual, ideal.

All current theories of spacetime that are more grounded than speculative (e.g. LQG isnt mainstream right now) require space and time to be infinitely divisible and no observation contradicts this at all. In fact, attempting to make those finite will result in inconsistencies more than likely. So no, observation does not require one to class the infinite as merely potential, a mere idea that cannot be found in the world. — MindForged

Spacetime is conceptual. This is the problem I had with your last post, you reified "space", making it into some sort of an object to justify your position. In reality, "space" is purely conceptual. We do not sense space at all, anywhere, it is a constructed concept which helps us to understand the world we live in. Furthermore, "infinitely divisible" is an imaginary activity, purely conceptual. We never observe anything being infinitely divided, we simply assume, in our minds, that something has the potential to be thus divided.

And in any case, the way you're talking about potentiality sounds contradictory. It's being spoken of as if it's ineffable. And yet you're telling me about it and what makes it ineffable... Which means youre talking about it, so it's not ineffable. — MindForged

I never defined "potentiality" as ineffable. It may appear to you that potentiality is contradictory ifyou do not understand the concept, but Aristotle was very specific and explicit in his description of what the term refers to, to ensure that his conception does not defy the law of non-contradiction. As that which may or may not be, he allowed potentiality to defy the law of excluded middle. You will find that some modern philosophies though, such as dialectical materialism, and dialetheism allow for violation of the law of non-contradiction. They may conceive of potentiality, or matter as contradictory.

This sounds incredibly wacky. For one, even if there is some metaphysical violation of Excluded Middle, that doesn't preclude it from human understanding nor does that make reality "violate the laws of logic" because there are not "the" laws of logic. There are many such sets of laws, and some drop Excluded Middle. It would certainly be a surprise to the Intuitionists that they don't understand constructive mathematics or their own logic because it doesn't assume EM as an axiom. — MindForged

it's not "incredibly whacky" it's the central point of our discussion. There are always things which escape human understanding. They are things which human beings do not understand. I would place "infinite" in that category, but you seem to think that "infinite" is understood, so let's go back four or five hundred years and say that at that time, "infinite" was not understood. The reason why it could not be understood was that it appeared to defy the law of non-contradiction, in the form of paradoxes. When some premises lead to conclusions which contradict what we know as basic fact, then there is a problem in our understanding. These are things which escape human understanding.

When this problem occurs, there are two principal possibilities of the cause of the problem. One is that the fundamental premises, how the terms are defined, are faulty, the other is that the fault is within the logical process. So our subject is "infinite", a few hundred years ago,when it escaped human understanding. The term produced problems which caused the appearance that it could not be understood. The logicians at the time decided that the best way to proceed was to change the premises, the defining terms of "infinite". What I am arguing is that misunderstanding is not due to faulty premises, but to faulty logical process. Zeno's paradoxes deceive the logician through means such as ambiguity or equivocation, by failing to properly differentiate between whether the aspects of reality referred to by the words, have actual, or potential existence. That's what Aristotle argued. So the logician gets confused by a conflation of actual problems and potential problems, which require different types of logic to resolve, and are resolved in different ways. Instead of disentangling the potential from the actual, the logicians took the easy route, which was to redefine the premises. All this does is to bury the problem deeper in a mass of confusion.

And this is all besides the point anyway. The nonsense you tried to pass off earlier was the idea that there are "defining features" of things like parallel lines despite now knowing that these terms are defined by the user's (implicitly or explicitly). They don't have inherent definitions, they're defined within a certain domain. So the idea that you tried to push that parallel lines don't intersect is purely based off the underlying assumptions of the geometry in which you made an assumption of, it is not true writ large in geometry. The Parallel Postulate is only true inasmuch as it's assumed to be so in a geometry and anyone saying otherwise is just misinformed about how mathematical formalisms work. — MindForged

You haven't addressed the issue here. You only support these claims with a reified "space", assuming that space is a physical object to be studied, and not a conceptual object. It's beginning to appear like this is the crux of the differences which we have. Do you distinguish between objects which exist solely in the mind, imaginary things, concepts and ideas, and things which have physical existence in the world, like rocks and trees, and planets? If so, then when we use words to refer to things, we ought not confuse whether the thing referred to is ideal, a concept, or a physical object. An infinity, just like infinite, is always an ideal, a concept, never something in the physical world.

I haven't reified anything, I didn't treat these as anything other than abstract mathematical constructs. — MindForged

What's this then?

Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). — MindForged

See, you are treating "space" as if it is something described by geometry. In reality, since we can use various different geometries to describe the various types of objects we sense, there is no such thing as "space". We might be able produce a concept of "space" from this geometry, and another concept of "space" from this other geometry, but it really makes no sense to talk about "how space is", or "if space is curved...", because there is no such thing as "space", not even as a concept.

This is why your geometrical examples are irrelevant, and way off the mark. You are talking about geometry as if it is created to describe some sort of "space". Then you need to bring in some principles to account for the intuition that space is in some sense infinite. However, this is totally uncalled for. We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects. It is only when we approach these concepts of geometry attempting to synthesize a concept of "space", that we find "infinite" as inherent within the principles themselves. Then there is a problem because we want to produce a concept of space which allows for the infinite because we have been confronted with the infinite as inherent within the concepts used to describe objects.. But this is only a pseudo problem. The infinite is not part of the objects which we measure and therefore it ought not be part of any concept of space. The infinite is only a part of the mathematical tools which we use to measure with. So any attempt to bring the infinite into our conception of "space" as a the thing being measured is a mistake.

Moreover, there is no ultimate compatibility between a continuum and the numbers--or anything else discrete — aletheist

That's the point, they are incompatible. And, the introduction of infinitesimals does not make them compatible. It's only an illusion.

The only reason for positing infinitesimals (in contrast to points) is to preserve those basic, primitive intuitions of continuity, rather than resorting to (wrongly) treating a continuum as if it were composed of discrete units. — aletheist

The problem with points is that a point is by definition, dimensionless. It takes up no space on a line, only divides a line. Therefore the line cannot be made up of points. Even an infinity of dimensionless points cannot account for the existence of the continuous, one dimensional line, which is supposed to exist between two points. There is a fundamental incompatibility between the point which is dimensionless, and the line, which is dimensional. Infinitesimal points were introduced to allow that a multitude of infinitesimals may have dimensionality, therefore a continuous line could be conceived of as being composed of infinitesimal points. This is an attempt to establish compatibility between the non-dimensional point, and the dimensional line.

Following a rule does not mean that one must follow it, but rather that one knows how to follow it. — Fooloso4

Do you not recognize that as blatantly contradictory?

In this case it is not a matter of whether or not there is such a rule. “Fetch a red apple” is the rule. — Fooloso4

How is that the rule? "Fetch a red apple" is the statement, what is said. If there is a corresponding rule, the rule would tell me what to do when I hear that statement. The rule cannot be the statement itself, because the rule must indicate what the correct action is when the statement is heard.

That was a direct quote from the text (§6). What problems are there? — Fooloso4

You should read the thread from the beginning, we covered that already.

The object serves as a paradigm. It is something that serves as an example of what red means. If someone does not know what red means I cannot tell them to remember what it means or to look in my memory. The particular object I point to can be destroyed but there are others that can serve as the paradigm. — Fooloso4

Wittgenstein explicitly states that the name must have meaning even if everything is destroyed. So it is clear that he is not thinking about a multitude of objects acting as the paradigm.

What cannot be destroyed is what gives the words their meaning, it “is that without which they would have no meaning” (§55) A paradigm can be destroyed but then word would no longer mean anything . If you were the last remaining member of a tribe and everything owned by the tribe was destroyed a “rel” would mean something to you but not to anyone else. Since no “rels” exist there is nothing that can serve as a paradigm by which “rel” means anything for anyone else, and if you forget or die then it would no longer have any meaning. The paradigm would be destroyed. — Fooloso4

You are misrepresenting what Wittgenstein says at 55. He explicitly says that the name would have meaning even if all the corresponding objects were destroyed. Then he says "An example of something corresponding to the name, and without which it would have no meaning, is a paradigm that is used in connexion with the name in the language-game." So we can conclude that "a paradigm" does not refer to any object, or a multiplicity of objects.
.

...a multi-billion dollar moron useless erection... — Hanover

How would you define "useless erection"?

No, infinitesimals are not units, and they are not "distinct things." — aletheist

You seem to have little understanding of what "infinitesimals" refers to. An infinitesimal is defined by Leibniz as an entity, a unity. It was used as an approach to mathematizing space, time, and matter which were previously conceived of as continuities. There was an inconsistency between mathematics which deals with distinct units, and these concepts, space, time, and matter, which were based in an assumption of continuity. If these continuities, space, time, and matter, could be conceived of as composed of infinitesimals (units, like monads) we could establish compatibility between a continuum and the numbers..

Peirce did nothing to change the fundamental nature of infinitesimals as units. The problem of course, is that if the things which we knew of as continuous, space, time, and matter, are really composed of these units, infinitesimals, then they are not truly continuous. So the question is whether these basic, primitive intuitions which hold space, time, and matter as continuous are correct, or are these things which appear to be continuous, more appropriately represented by the discrete units, infinitesimals. If the latter is the truth then we ought to be able to distinguish real boundaries between one infinitesimal and another. It does not suffice to say that the boundaries are there, but they are vague and cannot be determined. To give reality to the infinitesimals we need to determine those boundaries.

If you fetch a red apple whenever you are asked to then you know the rule. It is as simple as that. Fetching the apple is sufficient. What more do you think needs to be added? What is missing? Whether or not one is following the rule is determined by an action: — Fooloso4

It's not "as simple as that" though. Right now if someone said "fetch a red apple", I would not be in the least bit inclined to go to the store and get a red apple. However, if there is a rule involved with this phrase, I must still know the rule because I understand the phrase. However, I chose not to act according to the assumed rule. So despite the fact that acting in a certain way may indicate that I know the rule, if there is such a rule, it's not a reliable way of indicating whether I know that rule. And so it doesn't suffice as a premise, whereby we could conclude the existence of such a rule, because in each and every particular case when someone says "fetch a red apple", people behave differently.

“We could imagine that the language of §2 was the whole language of A and B; even the whole language of a tribe. The children are brought up to perform these actions, to use these words as they do so, and to react in this way to the words of others.” (§6) — Fooloso4

As Wittgenstein demonstrates, there are problems with this imaginary scenario of yours.

The paradigm is the highest court.

"An example of something corresponding to the name, and without which it would have no meaning, is a paradigm that is used in connexion with the name in the language-game." (§55)

A physical example is in general a more reliable paradigm provided it does not change. In addition, we are able to compare the sample with the name. I do not need to consult a sample of the color red each time I fetch a red apple, but if you fetch a yellow apple and say that this is how you remember the color red, then we can consult the sample to settle the matter of what red means. — Fooloso4

You seem to missing the fact that it is made explicitly clear by Wittgenstein at 55, that this so-called "paradigm" cannot be a physical object, because the name must be allowed to have meaning after the physical object is destroyed. What the name signifies must be indestructible, a physical example is not. Therefore a physical example is not what Wittgenstein refers to here as "a paradigm".

There are two fundamental mistakes here: first, infinitesimals are not units; second, a continuum is not composed of infinitesimals. — aletheist

Yes, infinitesimals are units, they are bounded necessarily in order to give them the status of distinct things "infinitesimals". without separation between them, there could not be the plurality indicated by "infinitesimals". Under Peirce's philosophy though, the boundaries are vague. However, it is necessary that either the boundaries are real in order that the infinitesimals are real, or else the boundaries are not real, in which case neither are the infinitesimals.

It violates the laws of classical (bivalent) logic, but that is not the only kind of logic available to us. For example, we can reason without the law of excluded middle using intuitionist logic. In fact, the law of excluded middle does not apply to infinitesimals; rather than discrete points, they are analogous to indefinite "neighborhoods" with an inexhaustible supply of potential points. — aletheist

Right, but doing this only gives "potential" a different definition, just like mathematics gives "infinite" a different definition. Producing a different definition only means that the corresponding object is not the same object. So we are no longer talking about what Aristotle identified as "potential" we are talking about something different.